Discrete Math

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Use mathematical induction to prove the truth of each of the following assertions for all n ≥1.

n³ + 5n is divisible by 6

I really do not understand this to much. This is what I have so far:

n = 1, 1³ - 5(1) = 6, which is divisible by 6

Then I really don't know how to go on from here. I appreciate any helpful replies. Thank you!

  • Discrete Math -

    The next step is to assume the proposition is true for n.
    The task is to show that if the proposition is true for n, then it would be true for n+1. Once that is established, then the proof is complete.

    6|n^3+5n => 6|1+5 is true for n=1
    Assume 6|N^3+5 is true for n, then
    for n+1
    (n+1)^3+5(n+1)
    =n^3+5n +3n^2+3n+6
    =n^3+5n +3n(n+1) + 6
    We now examine the three terms:
    n^3+5n is divisible by 6 by initial assumption.
    6 is divisible by 6.
    3n(n+1) falls into two cases:
    1. n is odd, then n+1 is even, therefore 6 divides 3*(n+1)
    2. n is even, then 6 divides 3n.
    Since all three terms are divisible by 6, we only have to extract the factor of 6 from each term and declare the expression (n+1)^3+5(n+1) is also divisible by 6.
    By the principle of mathematical induction, the proposition is proved. QED.

  • Discrete Math -

    Thank you so much for your response! But I have completed that particular question. However, can you please help with this one? I am confused. . .

    Use mathematical induction to establish the following formula.

    n
    Σ i² / [(2i-1)(2i+1)] = n(n+1) / 2(2n+1)
    i=1

    Thanks for any helpful replies :)

  • Discrete Math -

    Any suggestions?

  • Discrete Math -

    There are three steps:
    1. Basis:
    Test case for n=1 (or any other finite number):

    n
    Σ i² / [(2i-1)(2i+1)] = n(n+1) / 2(2n+1)
    i=1

    for n=1,
    Left hand side=1/[(2*1-1)(2*1+2)=1/3
    Right hand side=1(1+1)/[2(2*1+1)]=1/3

    So formula is established for n=1.

    2. Assume
    formula is valid for case n.

    3. Show that formula is valid for case n+1.

    Left hand side:
    n+1
    Σ i² / [(2i-1)(2i+1)]
    i=1

    =

    n
    Σ i² / [(2i-1)(2i+1)]
    i=1

    + (n+1)²/[(2*(n+1)-1)(2*(n+1)+2)]

    =

    n(n+1) / 2(2n+1) + (n+1)²/[(2*(n+1)-1)(2*(n+1)+2)]

    =

    [(n^2+n)*(2n+3)+2(n+1)^2] / [2(2n+1)(2n+3)]

    =(n+1)(n+2)(2n+1) / [2(2n+1)(2n+3)]
    =(n+1)((n+1)+1))/[2(2(n+1)+1)]
    =m(m+1)/[2(2m+1)]

    Which is precise the right hand side with m replacing n+1.
    QED

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